Stability of ODE with colored noise forcing.

I will discuss recent work on the stability of linear
equations under
parametric forcing by colored noise. The noises considered are built from
Ornstein-Uhlenbeck vector processes. Stability of the solutions is
determined by the boundedness of their second moments. Our
approach uses the Fokker-Planck equation and the associated PDE
for the marginal moments to determine the growth rate of the
moments. This leads to an eigenvalue problem, which is solved
using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes
into "ladder operators." The results are given in terms of a perturbation
expansion in the size of the noise. We have found very good
agreement between our results and numerical simulations. This is
joint work with L.A. Romero.